From particle mechanics to micromorphic media. Part I: Homogenisation of discrete interactions towards stress quantities

Ehlers, Wolfgang; Bidier, Sami

doi:10.1016/j.ijsolstr.2018.08.013

Cited by 10 publications

(8 citation statements)

References 52 publications

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“…where τ 1,2 are linear independent tangent vectors to ∂Ω. One then observes that the minimal energy content of a solution (u, P ) to the minimization problem (19), (2), (20), (21) is easily bounded above by choosing the macroscopic fields (u, P ) such that…”

Section: Maximal Stiffness On the Micro-scalementioning

confidence: 99%

“…where C(ξ) is the elasticity tensor of the aluminum phase or air depending on the position of ξ in the unit-cell 19 and E = sym∇u (x) is the applied straining at the macroscopic point x, where the unit-cell V (x) is centered at x. For the computation of these macroscopic elasticity coefficients we use the two-scale finite element method FE-HMM [20] (see also [53,77,78]) and we assume that the microproblem is driven under macroscopic plane strain conditions.…”

Section: Determination Of C Macro − Classical Periodic Homogenizationmentioning

confidence: 99%

“…Step a) seems to be uncritical, since most authors working in the field of extended continuum mechanics agree on the same or similar format of balance relations [19,21,22,39,40,54].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Neff

Eidel

d’Agostino

et al. 2019

J Elast

103

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We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann's principle on the overall representation of anisotropy. We explain our idea of "maximal" stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper [16] to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

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Section: Maximal Stiffness On the Micro-scalementioning

confidence: 99%

Section: Determination Of C Macro − Classical Periodic Homogenizationmentioning

confidence: 99%

See 1 more Smart Citation

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Neff

Eidel

d’Agostino

et al. 2019

J Elast

103

View full text Add to dashboard Cite

show abstract

“…The internal length scale parameters can be used to define size of the representative volume element (RVE) in homogenisation approaches controlling the non-local material behaviour [1,50,29,45,7]. In granular mechanics, the micromorphic continuum formulation has been shown to be able to address scale-dependent material behaviour as linked to an independent micro-kinematics of interacting particles [17,24,58]. It is also capable to deal with mass flux in growth and remodelling of biological material [31].…”

Section: Introductionmentioning

confidence: 99%

Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibers

Skatulla¹,

Sansour²,

Limbert³

2021

Preprint

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There has been increasing experimental evidence of non-affine elastic deformation mechanisms in biological soft tissues. These observations call for novel constitutive models which are able to describe the dominant underlying micro-structural kinematics aspects, in particular relative motion characteristics of different phases. This paper proposes a flexible and modular framework based on a micromorphic continuum encompassing matrix and fiber phases. It features in addition to the displacement field so-called director fields which can independently deform and intrinsically carry orientational information. Accordingly, the fibrous constituents can be naturally associated with the micromorphic directors and their non-affine motion within the bulk material can be efficiently captured. Furthermore, constitutive relations can be formulated based on kinematics quantities specifically linked to the material response of the matrix, the fibres and their mutual interactions. Associated stress quantities are naturally derived from a micromorphic variational principle featuring dedicated governing equations for displacement and director

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“…A leading method for homogenization in continuous materials is still the traditional finite element method (FEM), due to accessibility of academic or commercial software and the relatively small modeling error, where 2D and 3D first or second order finite elements play a crucial role. Other methods are also considered, such as some meshfree formulations [11] , fast Fourier transform (FFT)-based methods [12], or even the discrete element method [13] , which seems to be perfect for the cases where reinforcing or filling particles are densely packed into the matrix. There also exist plenty of various approaches for reducing the effort of FEM computations at the expense of accuracy.…”

Section: Introductionmentioning

confidence: 99%

Hysteretic Behavior of Random Particulate Composites by the Stochastic Finite Element Method

Sokołowski

Kamiński

2019

Materials

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Hysteretic behavior of random particulate composite was analyzed using the stochastic finite element method and three independent probabilistic formulations, i.e., generalized iterative stochastic perturbation technique of the tenth order, Monte-Carlo simulation, and semi-analytical method. This study was based on computational homogenization of the representative volume element (RVE), and its main focus was to demonstrate an influence of random stress in constitutive relation to the matrix on the deformation energies stored in the effective (homogenized) medium. This was done numerically for an increasing uncertainty of random matrix admissible stress with a Gaussian probability density function, for which the relations to the energies of the entire composite were approximated via the weighted least squares method algorithm. This composite was made of two phases, a hyper-elastic matrix exhibiting hysteretic behavior and a linear elastic spherical reinforcing particle located centrally in the RVE. The RVE was subjected to a cyclic stretch with an increasing amplitude, and computations of deformation energies were carried out using the finite element method system ABAQUS. A stress–strain history of the homogenized medium has been presented for the extreme and for the mean mechanical properties of the matrix to illustrate the random hysteresis of the given composite. The first four probabilistic moments and coefficients of the RVE deformation energy were determined and have been presented in addition to the input statistical scattering of the admissible stresses.

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From particle mechanics to micromorphic media. Part I: Homogenisation of discrete interactions towards stress quantities

Cited by 10 publications

References 52 publications

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibers

Hysteretic Behavior of Random Particulate Composites by the Stochastic Finite Element Method

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